Well-placement optimisation using sequential artificial neural networks

Abstract

In this study, a new algorithm is proposed by employing artificial neural networks in a sequential manner, termed the sequential artificial neural network, to obtain a global solution for optimizing the drilling location of oil or gas reservoirs. The developed sequential artificial neural network is used to successively narrow the search space to efficiently obtain the global solution. When training each artificial neural network, pre-defined amount of data within the new search space are added to the training dataset to improve the estimation performance. When the size of the search space meets a stopping criterion, reservoir simulations are performed for data in the search space, and a global solution is determined among the simulation results. The proposed method was applied to optimise a horizontal well placement in a coalbed methane reservoir. The results show a superior performance in optimisation while significantly reducing the number of simulations compared to the particle-swarm optimisation algorithm.

Keywords

Artificial neural network sequential artificial neural network global solution well placement search space optimization

Introduction

Well placement is one of the most important steps in conventional or unconventional field development. Reservoir simulation has been frequently used to determine the optimal locations in well-placement problems. The ideal method is to conduct simulations for all possible drilling locations, called as the exhaustive simulation method, to obtain a true global solution wherein the cumulative production or the economic value is a maximum. However, because of the time and cost required for simulation, various optimisation techniques such as the gradient-based optimisation and stochastic-search algorithm have been developed to reduce the use of reservoir simulation. In the optimisation techniques, the cumulative production of hydrocarbon or the economic value of a reservoir is set as an objective function, and a global solution is searched to maximize the objective function. In case that the objective function is set to negative, the optimisation techniques search for a minimum.

The gradient-based optimisation has been used to obtain the optimum well placement and well trajectory for horizontal wells (Bangerth et al., 2006; Forouzanfar et al., 2010; Sarma and Chen, 2008; Wang et al., 2007). The main advantage of the gradient-based optimisation is that optimal solutions could be found quickly. However, one of the shortcomings of the method is that it depends strongly on the initial guess; hence, it may be trapped in a local optimum because of the initial guess.

To overcome the drawback of the gradient-based methods, the following stochastic-search algorithms have been proposed: simulated annealing (SA), genetic algorithm (GA), particle-swarm optimisation (PSO), and imperialist-competitive algorithm (ICA). The stochastic algorithms are more robust with higher chances of obtaining a global solution compared to the gradient-based methods. By optimizing the schedule and location of horizontal wells with fixed orientations in oilfields, Beckner and Song (1995) first applied the SA algorithm to maximise the net present value (NPV). Norrena and Deutsch (2002) also applied the SA for optimizing the well placement. The GA, which is one of the most popular optimisation algorithms, has been used in well-placement problems (Bittencourt and Horne, 1997; Emerick et al., 2009; Lee et al., 2009; Salmachi et al., 2013; Yeten et al., 2002).

Onwunalu and Durlofsky (2010) applied the PSO for optimizing the well placement and found that the PSO helped in achieving a better performance compared to the GA. Feng et al. (2012) optimised the well placement in a coalbed methane (CBM) reservoir by integrating the reservoir simulation and PSO algorithm. Al Dossary and Nasrabadi (2015) applied the ICA for the first time in oil and gas industry to obtain the optimum well location for maximising the well productivity.

Although the stochastic optimisation method requires fewer runs of reservoir simulation than the exhaustive simulation method, the reservoir simulation has to be performed a significant number of times to calculate the objective function. To reduce further the simulation runs in the stochastic optimisation method, the surrogate model technique has been adopted. In particular, ANNs are widely used to predict the output values of reservoir simulations, which comprise various input data, and provide an optimal prediction through learning techniques based on the nonlinearity between the input and output data (Centilmen et al., 1999; Doraisamy et al., 1998; Foroud et al., 2012; Lee et al., 2011; Min et al., 2011; Shahkarami et al., 2014; Zameer et al., 2017).

Doraisamy et al. (1998) applied ANNs for rectangular, L-shaped, and irregular shapes of reservoirs. They employed the infill well location and distance between the existing wells and an infill well as input data and the cumulative production of each well as output data. Centilmen et al. (1999) employed geological characteristics as the input data for the ANN to optimise the well locations in various gas reservoirs. Min et al. (2011) estimated the production rate using the production-potential map, which reflects the relative productivity of each grid in the reservoir.

The stochastic-search techniques have the following limitations. First, the search process in the stochastic-search techniques is iterative. The objective function is repeatedly calculated for various cases within the search space to minimise or maximise its value. When the global solution is reached for the first time, the solution is perceived as one of the local optima. It is after a sufficient number of iterations that the solution is confirmed as the global solution. During these iterations, a number of reservoir simulations are required in addition, which makes the techniques require cost and time. Even a local optimum could be accepted as the global solution, when there is no change in the obtained optimal solution during a sufficient number of iterations.

Second, in case of using the ANN to minimise the number of simulations, the predicted value using the ANN deviates from the true value obtained in the simulation run. This implies that the optimal solution identified using the ANN might not be the true global solution. To compensate for the deviation in the result of the ANN, the global solution could be determined by performing reservoir simulations on a group of near-optimal solutions resulting from the ANN. However, it cannot be ensured that the true global optimum is always included in the group of the near-optimal solutions.

In this study, a new methodology is proposed to overcome the aforementioned limitations by employing a series of ANNs sequentially. In the proposed method, each ANN is trained by successively increasing the amount of training data, and the search spaces gradually decrease to the global optimum. The proposed method is applied to the well-placement problem in a CBM reservoir.

Artificial neural network

The ANN is one of the most popular and widely used methods. The method is inspired by the intent of having machines that can emulate the human brain. The ANN is a robust approach used to approximate discrete or continuous target values and has been applied in several applications such as pattern recognition, classification, clustering, time-series forecasting, function approximation, optimisation, signal processing, telecommunications, and robotics.

The ANN comprises an input layer, a hidden layer, and an output layer (Figure 1). The training input pattern is transferred to the input layer, and the input pattern is transferred layer by layer until the output pattern is generated in the output layer. The weight of a neuron represents a hidden characteristic in an input pattern. If the output pattern is different from the target pattern, the error is calculated and is propagated backward along the neural network from the output layer to the input layer. The weights are corrected as the error is corrected. The optimal number of nodes in the hidden layer varies with the size of the optimisation problem. A guideline for the number of nodes in the hidden layer is expressed in equation (1).

N_{hidden} \geq \sqrt{N_{input} \cdot N_{output}}

(1)

where

N_{hidden}

N_{input}

, and

N_{output}

are the number of hidden, input, and output nodes, respectively (Masters, 1993).

Figure 1.

Schematic of artificial neural network.

Methodology

The proposed sequential ANN method is a method wherein the search space is gradually reduced by sequentially applying a series of ANNs and selecting a search space wherein the objective functions satisfy the predefined criteria. The amount of training data required for subsequent ANNs increases, which comprise the training data used for the preceding network and a certain amount of data selected within the new search space determined by the preceding network. The prediction performance could be improved by repeatedly constructing an ANN with accumulation of training data. When the search space size satisfies the predefined stopping criterion, the reservoir simulations are performed for all cases in the remaining search space and the true global solution can be found.

Figure 2 depicts the flowchart of the sequential ANN method. The detailed procedure is described as follows:

The initial training data are obtained through the reservoir simulation and are used to train an initial ANN model. It is sufficient for the number of the initial training data points to be 10 or 20, but it depends on the size of the problem.

After defining the search space for obtaining the global optimum, objective functions are predicted for the data in the search space using the ANN model. The objective functions can be set as the cumulative oil or gas production, NPV, or both.

The objective functions, including the values of the training data, are sorted in descending or ascending order, and the top R ranks of the objective functions are selected, where R is the cutoff criterion. The corresponding input data are designated as a new search space for the subsequent ANN. The probability of the global solution located in high rank gets lower and lower as the process goes on, because the search space gets more and more saturated with the near-optimal solutions and ANN model has its intrinsic error in prediction. This problem can be avoided by set larger value for the cutoff as the process goes on.

To train the subsequent ANN model, N data points newly chosen from the new search space are added to the training set, where N is equal to or smaller than the number of the initial training data points. The reservoir simulations for N data points are performed to add the input–output pairs as the training data. Increase the size of the training dataset can improve the prediction performance of the trained ANN model.

The subsequent ANN model is used to estimate the objective functions in the new search space.

To reduce the search space until the stopping criteria is satisfied, step 3 through step 5 is repeated. One of the stopping criteria can be defined as follows: Let the reduced search space T be defined as the number of data points that are not used as the training data in the new search space. When T is equal to or less than 2 times N, the stopping criterion is satisfied. When $T \leq 2 N$ , reservoir simulations are conducted for the remaining data in the search space instead of repeating one more iteration. It is reasonable to stop the procedure, because the new search space is more saturated with near-optimal solutions and ANN generally has intrinsic error in prediction. The global solution could be found as the maximum or minimum in the objective function among the simulation results from the T data and the training data.

Figure 2.

Flowchart of sequential artificial neural network method.

Results and discussion

Application to CBM reservoir

The proposed sequential ANN method is applied to the optimal well location problem in a CBM reservoir. The CBM reservoir is producing methane gas for three years through six vertical producers (p1–p6) as shown in Figure 3. The target is to obtain the optimal location of an infill horizontal well to maximise field production for 20 years. The reservoir has dimensions of 6.1 km × 3.7 km comprising 6771 grids with dimensions of 61 × 37 × 3. The length of each grid is 100 m in x and y directions. The reservoir thickness is in the range of 3–10 m. Table 1 lists the properties of the reservoir.

Figure 3.

Sector model of CBM reservoir.

Table 1.

Properties of CBM reservoir.

Reservoir parameters	Values
Cleat spacing	10 cm
Sorption time	1 day
Reservoir pressure	5456 kPa at 300 m
Gas type	CH₄
Langmuir volume; V_L (CH₄)	0.14–0.74 gmol/kg
Langmuir pressure; P_L (CH₄)	2887 kPa
Permeability model	Palmer & Mansoori
Matrix permeability	0.001 md
Permeability of face cleat	44–105 md
Permeability of butt cleat	11–26 md
Matrix porosity	0.08
Cleat porosity	0.02–0.03

CBM: coalbed methane.

The horizontal section of the infill well is set along the east–west or north–south direction. The horizontal interval of the infill well is assumed 300 m located at the centre of the reservoir thickness. Moreover, it is assumed that the well section cannot be located within the boundary grids and grids next to the existing well grids. The total number of cases wherein the infill well could be located in the reservoir is 3550. To verify the sequential ANN method, 3550 reservoir simulations were performed, and the global solution was identified in advance. The simulations were conducted using GEM developed by the Computer Modelling Group Ltd. (CMG, 2016).

Design of sequential artificial neural networks

The ANN model consists of one input, one hidden, and one output layer. The number of nodes in the input, hidden, and output layer are 15, 10, and 1, respectively. The input data used in the input layer of the ANN model are the heel and toe locations of the infill well, direction of the infill well, distance between the existing wells and the infill well, and distances between the infill well and the reservoir boundaries (Table 2). The output data used in the output layer are set as the 20-year cumulative production of methane gas in the reservoir.

Table 2.

Input data for artificial neural network.

Input data	Data type	# of nodes
Infill well heel x–y coordinates	Spatial information of well	2
Infill well toe x–y coordinates	Spatial information of well	2
Horizontal direction of the infill well	Spatial information of well	1
Distance from the infill well to the reservoir boundary	Spatial information of well	4
Inter-distance between existing wells and the infill well	Well interaction data	6

The number of the initial training data points used to train the initial ANN model is 20 with 10 in the east–west direction and 10 in the south–north direction as shown in Figure 4. The red dot and the red line represent the heel location of the infill well and direction of the well, respectively. Among the training data, 85% of the input data were used for training and 15% for testing. The cutoff value R is set to 15, 20, and 30% for the initial, second, and third ANN models, respectively, and 40% for the fourth and subsequent models. To increase the likelihood of inclusion of the global solution in the search space, the cutoff was performed to the search space including all the existing training data. N is set to 10, which is the number of training data points newly added to the existing training data for the subsequent ANN model. The Bayesian regularisation algorithm is used for model learning. The ANN model was trained using MATLAB of MathWorks, Inc. (MathWorks, 2016).

Figure 4.

Well locations for initial training data.

Global solution of infill-well placement

The initial ANN model was trained using 20 training data points. Figure 5(a) shows the training result, indicating that the reliability of the model is considerably high. The cumulative gas production is estimated for 3550 cases of the search space using the ANN model. Figure 5(b) shows the plotted result, where the horizontal and vertical axes represent the simulation and ANN results, respectively. To compare the model prediction with the simulation results, both the ANN and simulation results are cross-plotted. The light green points indicate the training data used to train the initial ANN model, and the blue circles indicate the estimation performed using the model. The red circle indicates the true global solution, which means that the gas production is maximised from the simulation result. The pink horizontal line is referred to as the cutoff value R of 15%; thus, the points above the line belong to the top 15% in terms of the ANN estimation. The search space is reduced to the top 15%, i.e. 533 cases, wherein 2 training data points are included as shown in Figure 5(b). Consequently, the new search space, of which the size is 551, is determined by the union of the top 15% cases and the training data. It becomes the search space for the next ANN model. The reduced search space T = 531, which is the size of the new search space excluding the training data. The global solution is at the top 8.42%.

Figure 5.

(a) Training result of initial ANN model, (b) cross plot of ANN model and simulation results, and (c) top 15% of search space using initial ANN model.

Figure 5(c) shows the new search-space map. The light green rectangles represent the training data, and the red rectangles represent the reduced search space T. The blue rectangles represent six existing wells. The yellow rectangle represents the global solution and is included within the new search space. A rectangle in Figure 5(c) represents only the heel location of the horizontal interval of a well, and the well direction is not indicated because of the complexity.

To train the subsequent ANN model, 10 data points from the reduced search space T in Figure 5(c) are randomly selected and are incorporated into the training dataset after performing the simulation. Figure 6(a) shows the training result of the 2nd ANN model using 30 training data points. The training result is better than that of the initial ANN model. Figure 6(b) shows the cross plot of the ANN estimations and simulation results. In this case, the cutoff R is set to the top 20% of the ANN estimations. Using this cutoff value, the size of the new search space is 136, and the reduced search space T = 106. The global solution is at the top 8.89%. Figure 6(c) shows the resultant search-space map. Compared to the preceding search-space map in Figure 5(c), the search space is converging to the region near the global solution.

Figure 6.

(a) Training result of 2nd ANN model, (b) cross plot of ANN model and simulation results, and (c) top 20% of search space using 2nd ANN model.

In the next step, additional 10 points in T of Figure 6(c) are selected and are added to the training dataset after simulation. The 3rd ANN model is constructed using a total of 40 training data points as shown in Figure 7(a). A new search space is then selected with the cutoff value of 30%, as shown in Figure 7(b). Figure 7(c) shows the search-space map. The size of the new search space selected by the 3rd ANN model is 78, and T = 38. The global solution is at the top 2.21%.

Figure 7.

(a) Training result of 3rd ANN model, (b) cross plot of ANN model and simulation results, and (c) top 30% of search space using 3rd ANN model.

The 4th ANN model is trained using 50 training data points after randomly selecting 10 additional points from T in Figure 7(c). Figure 8 shows the training results and new search space with the cutoff value of 40%. The global solution is in the top 1.28%. After conducting four iterations of the algorithm, T = 20, which means that only 20 cases remain in the search space that are not used to the training data. It meets the stopping criterion. Hence, instead of performing further iterations, simulations for all 20 cases are performed. The maximum value, which means the global solution in this application, is found among the simulation results of the training data and the 20 cases in T.

Figure 8.

(a) Training result of 4th ANN model, (b) cross plot of ANN model and simulation results, and (c) top 40% of the search space using 4th ANN model.

The search-space maps in Figures 5(c) to 8(c) show that the search space is gradually narrowing down around the global solution. Moreover, the training data are added sequentially near the global solution, which is considered to help in improving the accuracy of the ANN models as in Figures 5(a) to 8(a). Table 3 summarises the result of the sequential ANN method applied to obtain the optimal well placement in the CBM reservoir. The global solution is successfully found after four ANN models are trained sequentially and 70 runs of simulation are performed. Figure 9 shows the optimal infill well location and the cumulative gas production of each well. The cumulative gas production of the field is 7.978 × 10⁸ m³ after 20 years production. The cumulative gas production of the infill well is 2.209 × 10⁸ m³, which corresponds to 27.7% of the field total.

Table 3.

Summary of applying the sequential ANN method to the CBM reservoir.

ANN model	Cutoff value, R(%)	# of training data	Size of search space^a	T^b	Rank of the global solution(%)	Total # of simulation runs
1st	15	20	551	531	8.42	20
2nd	20	30	136	106	8.89	30
3rd	30	40	78	38	2.21	40
4th	40	50	70	20	1.28	50
–	–	–	–		–	70 ^c

CBM: coalbed methane.

The size of the search space is the union of the top R rank data and the training data in each iteration.

The size of the search space excluding the training data.

Simulations for the 20 cases in T after the 4th ANN model are considered.

Figure 9.

(a) Optimal infill well location found using the sequential ANN and (b) cumulative gas productions of optimal infill well and existing wells.

The sequential ANN method was applied repeatedly for 10 times to investigate the reliability of the method. Table 4 summarises the results. The required iterations to train the ANN models are four for all cases. The total simulation runs are in the range of 65–70. The global solutions are located at very high ranks. Although the rank of the global solution of case 4, given in Table 4, is slightly low (14.39%) resulting from the 1st ANN model, the rank is improved as the process progresses.

Table 4.

Results of repeated application of sequential ANN method.

	Rank of global solution for each ANN model				Total # of simulation runs
Case	1st(%)	2nd(%)	3rd(%)	4th(%)	Total # of simulation runs
1	1.83	0.26	1.46	2.63	68
2	1.55	0.18	1.46	2.67	68
3	1.80	2.55	0.73	1.37	65
4	14.39	10.75	1.47	1.32	69
5	4.48	0.55	0.74	2.67	68
6	3.89	0.91	0.72	2.63	70
7	0.45	0.18	0.74	2.70	67
8	1.1	2.01	0.73	2.63	70
9	2.08	1.64	0.72	1.32	68
10	4.48	0.55	0.74	2.67	68

Comparison with PSO algorithm

The PSO is a stochastic optimisation algorithm proposed by Kennedy and Eberhardt (1995). The PSO was derived from the research on the social behaviours of bird foraging. The detailed description of the algorithm can be found in Appendix. The PSO algorithm was performed for the aforementioned CBM reservoir to compare with the sequential ANN method in terms of the number of simulation runs. After performing the sensitivity study on the parameters of the PSO, the following values were selected. The number of particles is set to 20, and the maximum number of iterations for the PSO algorithm is set to 20. The components of a particle consist of heel position of an infill well and direction of the horizontal section. The weights c₁ and c₂ for the velocity in equation (2) are 1.5 and 2.5, respectively. The objective function is set to negative by multiplying the cumulative gas production by minus one. The PSO algorithm was used to obtain the minimum value of the objective function as the global solution. In each iteration of the PSO algorithm, 20 particles move to new locations. Each particle represents each case of the infill well placement. Thus, 20 cases of the infill well placement are generated for each iteration. Reservoir simulation is required if a location is visited for the first time.

Figure 10 shows the trend in the minimum value of the objective function with respect to the number of iterations of the PSO algorithm conducted for the four repeated cases. In cases (a), (b), and (c), shown in Figure 10, the global solution was found as − 7.978 × 10⁸ that was the cumulative gas production multiplied by minus one. However, in case (d), the solution converged to a local minimum of − 7.756 × 10⁸ and the global solution could not be found. Table 5 presents the summary of the results. The simulation runs required for the algorithm to obtain the global solution for the first time for each case are 93, 96, and 125, respectively. However, this is not sufficient to confirm that the global solution has been reached at this point, because it is uncertain whether the obtained minimum value would change further. It can be confirmed that the global solution is obtained only when the minimum value has not changed for a sufficient number of iterations.

Figure 10.

Objective functions of PSO with respect to iterations for four repeated cases.

Table 5.

Summary of PSO results.

Case	Iterations at which the global solution is obtained for the first time	Simulation runs required for the iteration in the 2nd column	Simulation runs required for 20 iterations
(a)	8	93	124
(b)	13	96	131
(c)	15	125	157
(d)^a	–	–

Case (d) converges to a local minimum instead of the global solution.

In cases (a), (b), and (c), wherein 20 iterations of the PSO algorithm were performed, the required simulation runs were 124, 131, and 157, respectively (Table 5). It can be confirmed that the global solution was obtained after !!20 iterations in case (a) because the minimum of the objective function had not changed for the last 12 iterations as shown in Figure 10. However, the same is unsure for case (c) because of the slightly short duration (5 iterations) during which the minimum of the objective function is constant. Hence, more iterations may be needed to confirm the global solution for case (c). It should be noted that in case (d), the global solution could not be found despite the long duration of the constant value in the objective function. Compared to the PSO algorithm, the sequential ANN method, which requires at most 70 simulation runs for the same problem, is very efficient in terms of simulation runs required to obtain the global solution.

Conclusions

In this study, a sequential ANN method is proposed to obtain the optimal well placement. In the sequential ANN method, the search space is reduced while constructing the ANN sequentially. The ANNs are newly trained at each iteration using the training dataset that is added gradually with a predefined number of data points within the reduced search space, which improves the estimation performance of the network. As the iteration progresses, the cutoff value, above which the search space narrows down, is increased to reduce the possibility of the global solution dropping out of the subsequent search space.

When applying the sequential ANN method to a CBM reservoir, the ANNs were trained four times sequentially and were successfully used to find the optimal location of a horizontal infill well by performing 70 simulation runs. The result of repeated application to the same problem indicates that the sequential ANN method is reliable. When the PSO was used for the same problem, at least 124 simulation runs were required to recognise that the sought solution could be the global solution. Compared to the PSO, the sequential ANN method was verified as a more efficient algorithm. The performance of the sequential ANN method depends on its parameters. It is noted that the parameters of the sequential ANN method should be adjusted according to the reliability of the ANN training result and the size of the problem. The sequential ANN method can be applied to the problems where the optimal solution should be searched in a finite search space with the hard computing, e.g. an expensive and time-consuming reservoir simulation.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was supported by the Energy Efficiency & Resources program (No. 20152510101880) of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korea Government Ministry of Trade, Industry and Energy.

Appendix: PSO algorithm

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